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“Analysis and Design of Fuzzy Systems”
by Li-Xin Wang
June 1992
The goal of this thesis is to develop methods which can combine both numerical information and linguistic information into a common framework, and to apply them to a variety of control, signal processing, and communication problems. This goal is important because so much human knowledge is represented in natural language and to incorporate it into engineering systems is clearly needed. We show that fuzzy systems are powerful tools for achieving this goal. We first present a systematic description of fuzzy systems and show that the fuzzy systems are very general and include probabilistic general regression as a special case. To justify the use of fuzzy systems as basic building blocks for engineering systems (controllers, identifiers, predictors, filters, etc.), we rigorously prove, using the Stone-Weierstrass Theorem in mathematical analysis, that the fuzzy systems are capable of approximating any nonlinear function over a compact set to arbitrary accuracy. Then, three design methods for fuzzy systems are developed which determine fuzzy systems based on desired input-output pairs and fuzzy IF-THEN rules from human experts.
In the first method, we show that fuzzy systems can be represented as three-layer feedforward networks, and develop a back-propagation algorithm to train the fuzzy systems to match desired input-output pairs. We use this back-propagation fuzzy system as identifiers of nonlinear dynamic systems and show, through simulations, that the performance of the fuzzy identifiers is much better than the neural network identifiers. The second method is based on the classical orthogonal least squares algorithm; the basic idea is to represent the fuzzy systems as expansions of fuzzy basis functions and use the orthogonal least squares algorithm to select the significant fuzzy basis functions. We apply this method to the control of the nonlinear ball and beam system. The third method is a simple one-pass procedure which is based on a five-step procedure of generating fuzzy rules from numerical data. We apply this method to the truck backer-upper control and time-series prediction problems and show that the performance of this new method is better than that of the conventional fuzzy and neural approaches. Finally, we develop two nonlinear adaptive filters based on fuzzy system models, namely RLS and LMS fuzzy adaptive filters, and use them as equalizers for nonlinear communication channels. We show that the fuzzy adaptive filters are well-defined nonlinear adaptive filters, and have the unique advantage (as compared with polynomial, neural nets, and radial basis function, etc., adaptive filters) of directly incorporating linguistic information from human experts into the filters.